The spectral representations for isotropically correlated two l₁- and l₂-vecotr random fields are given generally in terms of l-vector harmonics and random measures, where l-vector denotes a (2l+l)- dimensional vector in the irreducible representation space of weight l of the rotation group, l＝0 being a scalar and l＝1 an ordinary vector. The representation is derived by making use of the multi-dimensional moving average and the previous work on l-vector functions. The three special cases of interest are discussed in detail ; namely, two scalar random fields, two vector random fields and a scalar and a vector random field

OGURA, Hisanao

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Simultaneous SpectralRepresentations of IsotropicallyCorrelated Random FieldsOGURA, HisanaoOGURA, Hisanao. Simultaneous Spectral Representations of Isotropically CorrelatedRandom Fields. Memoirs of the Faculty of Engineering, Kyoto University 1970, 31(4): 564-5751970-03-31http://hdl.handle.net/2433/280796564 Simultaneous Spectral Representations of Isotropically Correlated Random Fields By Hisanao 0GURA * (Received June 27, 1969) The spectral representations for isotropically correlated two l,- and /2-vecotr random fields are given generally in terms of /-vector harmonics and random measures, where /-vector denotes a (2/+l)- dimensional vector in the irreducible representation space of weight l of the rotation group, l=O being a scalar and l= I an ordinary vector. The representation is derived by making use of the multi-dimensional moving average and the previous work on I-vector functions. The three special cases of interest are discussed in detail; namely, two scalar random fields, two vector random fields and a scalar and a vector random field. I. Introduction In the foregoing papers [l], [2] (the latter will be referred to as I) the author has discussed the spectral representation of a homogeneous and isotropic scalar random field and that of vector random field. The representation is given in terms of solid scalar or solid vector harmonics and orthogonal random measures. In the present paper we concern ourselves with simultaneous spectral representations of random fields having isotropic correlations among them; for example, temperature and density fields of atmosphere, pressure and velocity fields of turbulent flow, electric and magnetic fields in the black-body radiation etc. We deal with two such fields simultaneously, but the case of three or more can be discussed analogously if necessary. As above examples, the following three cases are of interest to us from the point of view of physics and engineering; namely, the case of two scalar fields, that of a scalar and a vector field and that of two vector fields. To give a unified description of these cases, we will generally deal with [,-vector and [2-vector random fields simultaneously; [-vector means a (2l + 1 )-dimensional vector defined in the space D 1 of the irreducible representation of weight l of the 3-dimensional rotation group, a scalar corresponding to l=O and a vector to l= 1 [3]. Concerning * Department of Electronics Simultaneous Sepctral Representztions of Isotropimlly Correlated Random Fields 565 various definitions, notations and formulae related to the theory of rotation group and !-vector functions, we draw freely from a previous work of the author on the addition theorems of spherical Bessel functions and vector harmonics [ 4], to which we will refer as II. We will see that the 71 X /2-tensor addition formula of a gener-alized spherical Bessel function plays an important role in the isotropic correlation between the two random fields. As in the previous works, we derive the spectral representation in a somewhat formal manner from a multi-dimensional moving average; therefore we leave the general proof of the spectral representation to a future work. The main theorems of the present paper have been reported in ref [5] 2. Homogeneous [-dimensional random field Just asap-dimensional normal (Gaussian) random measure is defined in I, we define a p-dimensional random measure B(A) on a 3-dimensional space R3 , A denoting a set on R3 of finite measure m(A). Let eu denote sample point in the sample space Q, and E() the average over Q/ We often delete eu from notations for simplicity. A p-dimensional random measure B(A) = B(A, eu) = {B'"(A, eu), a= I, 2, ... ,p} (2.1) is a system of p random variables depending on a set A in such a way that (2.2) where ocr,fl denotes the Kronecker symbol. For mutually disjoint sets An, n= 1, 2, ···, we can write (2.3) which means (2.4) Then the 'complete additivity' holds in the sense of mean convergence; = = B ( ~ A,.) = ~ B (A,.) . n::..-::} n=1 We define a p-dimensional vector space Va in which, for any pair of vectors h={h'"} and g={g'"} (a= 1, 2, , .. , p), the scalar product p -(h, g) = ~ h'"g'" (2.5) Q',=1 and the square length (h, h) are defined. We denote by L 2 (R3) a Hilbert space of Vu-valued vector functions f(x)={f'"(x)EL2 (R3 ), a=l, •··, p}, with the inner product 566 Hisanao 0GURA (f, g) = ~R3 (f, g)dx = .t (j'», g .. ), and the square norm llfli2=(f, f), where defines the inner product in L2 (R3). (2.6) (2. 7) Asp-dimensional Wiener integral defined in I, we can define a p-dimensional stochastic integral off EL2 (R3) with respect to B(A): /(f) = ~R/(x)dB(x, w) = ,t LJ .. (x)dB .. (x, w), where we have put B(dx) =dB(x) and B .. (dx) =dB .. (x). I(f) properties: /(af+hg) = al(f) +hl(g), E(/(f)) = 0, E(/(f)/(g)) = (f, g), E(II(f)l 2) = 11£112. has (2.8) following (2.9) (2.10) (2.11) (2.12) A homogeneous [-dimensional random field I;(x), i=l, 2, •··, l, on R3 can be represented as a !-dimensional moving average in terms of l p-dimensional stochastic integrals off;(x)EL2 (R3), i=l, 2 ,···, l: I;(x) = f f;(x-x')dB(x') = ± f Jt(x-x')dB .. (x') , (2.13) .)R3 '"=l .)R3 which, by (2.11), has the correlation matrix As shown in J §3, the moving average (2.13) has the spectral representation I;(x) = f e2"'Cx,u)dM,(y, w), i = 1, 2, ···, l, h3 (2.15) where dM,(y) = M,(dy) and M,(S), S denoting a set on R3 , denotes the [-dimensional random spectral measure having properties E(M,(S)) = 0, E(M,(S)M;(S')) = f F,;(y)dy, .lsns' (2.16) (2.1 7) (2.18) The matrix-valued spectral density F;;(Y) is Hermitian non-negative definite and Simultaneous Spectral Representations of lsotropicaUy Correlated Random Fields 561 FHy) is the Fourier transform ofjf (x.): jf(x.) = f e2"icx,g) Ff (y)dy, F~(y) = FH -y) . JR3 The correlation matrix R;i(x) has the spectral representation with the inversion formula Since F;i(Y) is non-negative definite Hermitian matrix, we have A /-dimensional random field I;(x) is called degenerate if det (Fii) = 0 (2.19) (2.20) (2.21) (2.22) (2.23) holds for almost ally, and the largest rank of (F;;) for which the set ofy has non-zero measure is called the rank of the spectral density matrix, which turns out to be the degree of freedom of I;(x). For F;;(Y) given by (2.18), the rank r is equal to that of FHy). Hence, if p<l, (2.15) is always degenerate. If r<l, l-r functions from among l p-dimensional vector functions F';={F1, a=l, 2, •··, p} can be expressed as, say, and hence, where a;;(x) denotes the Fourier transform of A;i(y). Thus I;(x.) = t f a;;(x.-x.')l;(x.')dx.', i = r+l, ···, l; j=l JR3 (2.24) (2.25) (2.26) that is, when a homogeneous /-dimensional random field is degenerate with rank r, its l-r components can be derived from other r by means of a linear operation commuting with spatical translations. In what follows, we put l=p without loss of generality. 568 Hisanao 0GURA 3. Isotropically correlated two random fields In the following sections we consider /-vector random field defined in the representation space D 1 • A /1 -vector and a /2-vector random fields have (2/1 + 1) and (212+ 1) components respectively. When viewed differently, these two /-vectors together can be regarded as a (2/1 + 1) + (2/2 + I )-dimensional vector in the sum-space of representation D11 +D12 , which we call /1 +!2-vector for convenience. The correlation function of this random field, therefore, is a tensor in the product space The canonical component' of the correlation tensor in each subspace DAX D,,,(J.., µ = Li, /2) is defined by R£~(x.) = E(/cA)m(x.)lu,.Jn(0)) m= -J.., ... ,J.., n= -µ, .. •,µ, (3.2) The correlation tensors of /1 - and /2-vector random fields themselves are defined in in the first and the last space on the right-hand side of (3.1), while the mutual correlation tensors are defined in the second and the third space. When the mutual correlations are isotropic tensor fields2, then we say that the two random fields are isotropically correlated. In the following we deal with homogeneous and isotropic random fields which are isotropically correlated, and hence the correlation tensor of the /1 +!2-vector random field is an isotropic tensor field in the product space (3.1). Applying Lemmata 2 and 3 of II, Appendix II, to (2.20) and (2.21), we obtain the following result: THEO REM 1. ( Spectral representation of correlation tensor) For the homogeneous and isotropic /1- and /2-vector random fields which are isotropically correlated with each other, both the correlation tensor R~: and the corresponding spectral density tensor F~,,, are isotropic Jxµ-tensor fields in JJAxD,,. (J.., µ=li, l2). They are mn represented as 'diagonal' matrices in canonical components: 1 See II, Appendix I. R~1;(x.) = om,.R~IL(r) ' F~:(y) = DmnF~fL(t) ' r =Ix.I, t = IYI' (3.3) (3.4) 2 For definition of isotropic tensor field, see I, §4. General form of isotropic tensor field as well as its Fourier transform are given in II, Appendix II. Simultaneous Spectral Representations of Isotropically Correlated Random Fields 569 where F !"' is a non-negative definite Hermitian matrix with respect to the indices ,l, µ. R>.."' and p>--µ. satisfy the following symmetries R;t(r) = (-1)>..-,,.R~~(r) , F!"'(t) = F~>-(t), F!>-(t) ),0. (3.5) (3.6) The correlation tensor has the following spectral representation m terms of the generalized spherical Bessel function3 R;t(r) = 4n-.t\ [ j':,.~(2n-tr )F!"'(t)t2dt, m = -L, ... ,L with the inversion formula F~"'(t) = 4n-mtL [ j':,.~(2n-tr)R;t'(r)r2dr, n = -L, .. ,,L where L denotes the smaller of ,l and µ, (3. 7) 4 (3.8) The random fields have the following representations in terms of solid !-vector harmonics5 and random spectral measures. The derivation of the theorem will be given in section 5. THEOREM 2. (Spectral representation of random field) The above random fields have the following simultaneous spectral representations: >. oo / foo Ic>.i(r, e, <p) = v 4n-.~>,. ~S~/ Jo Ji~)n(2n-tr, (}, cp)dM[t)n(t, w), (3.9) ,l = !1 , !2 , where dM[t)n(t)=M/X)n(dt) and MUln(A) denotes the random spectral measure defined on the half-line T, 0,;;;; t < oo, having properties E(M/t)n(A)) = 0, E(Mtt)n(A)M/;f~,(A')) = Dnn1Dtt'Os/4n- ( F~"'(t)t2dt' J An A 1 (3.10) (3.11) A and A' being any intervals on T. 3 For definition, see II, §2.2. 4 According to II, (A. 46), the Fourier transform of a ;r xµ-tensor R>:"' is given by iµ.->.p~1•. mn mn However, we take F1~ as the spectral density tensor to supress the useless factor i >.-µ. without affecting the Hermitian property. For 2=µ=1 1=1, (3.7) reduces to I, (4.12) for a single vector random field. 5 For definition, see II, §3. 570 Hisanao OouRA The spectral representation of correlation tensor (3. 7) can be immediately obtained from (3.9): We multiply two expressions for arbitrary observation points and take the average using (3.11). We then recover (3.3), (3.7) on making use of the tensor addition theorem given by II, (4.1). In particular when J..=µ= l1 =l, (3.9) agrees with the formerly obtained result I, (5.17) for a single vector random field. 4. Examples of isotropically correlated random fields We write down the above obtained result for the three cases individually, which are of interest to us as mentioned in the introduction. 4.l. Scalar-Scalar (!1 =l2 =0). We descriminate the two scalar fields by the indices J.. =I, 2, instead of lu [2 both of which equal 0 in the present cases. There would be no confusion in doing this. The representation (3.9) can be written in terms of solid scalar harmonics6 ; l>..(r, 0, <p) = V 4ir ~ ,t, [ }15 (2irtr, 0, cp)dM/fi(t, cv), J.. = I, 2, ( 4.1) (4.2) (4.3) An example of two such scalar fields is furnished by the temperature and density fields of atmosphere which are statistically correlated with each other. The random fields in this case may be considered as nondegenerate. 4.2. Vector-Vector (!1 =12 = I). As above, the two vector random fields are labelled by J.. =I, 2. The corresponding spectral representation is given by (3.9) with !-vector harmonic J/tm replaced by vector harmonic J~•, n= -1, 0, 17• The isotropic correlation tensor R1~(x) = ommR~µ,(r), m, n = -1, 0, I, J.., µ=I, 2, (4.4) when expressed in the matrix with row and column numbered by n=-1, 0, I, corresponding to the canonical bases in D 1 +Du has the form 6 For definition, see II, (5.20). 7 See II, §5. l. Simultaneous Spectral Ripresentations of Isotropically Correlated Random Fields 571 Rll -1 '. Rl2 : -1 R22 -1 The corresponding tensor has the spectral representation (4.5) wherej:,.,,.=.j:,.1;.8. An example of isotropically correlated two vector random fields is given by the electric and magnetic fields in the black-body radiation. A vector random field 12 =rot 11 was already discussed in I, §6.3. When I, and 12 are con-sidered simultaneously, the corresponding spectral density tensor has the matrix form -211:t I F-1 I 2 0 (4.7) -211:tlF-11 2 l2n:tF_1l 2 0 0 2n:tlF,l 2 1211:tF,1 2 for which det IF mn I =0, showing the degeneracy of 6-dimensional random field I1+I2. 4.3. Vector-Scalar (l1 =l, l2=0). Representations (3.7) and (3.9) may be written down separately for two fields: I(r, fJ, <p) = y' 4n:nt, ~ St/ r J;'(2n:tr, fJ, <p)dM;'(t), I (r, fJ, <p) = y' 411: ~ :t, r ]'5 (211:tr, fJ, <p)dM15 (t) , E(Mf'(!:,,.)M;~''(!:,,.')) = onn'ou'os/4rc f ,I Fn(t) l 2 t2dt, .L.nt. E(M15 (!:,,.)M 1151 (!:,,.')) = o//'s/4n: r ,IF(t)i 2 t2dt, J t. nt. (4.8) (4.9) (4.10) ( 4.11) 8 For explicit representation, see II, (5.2). It is called the spherical Bessel vector function. Se also I, §4.2. 572 ltisanao OouRA ( 4.12) where we have put F1;1(t)=IFn(t)1 2 and F 22 (t)=IF(t)l 2 • Then nonvanishing components of the correlation tensor have the representations Rm(r) = 4nj:;j~ j;.,.(2n:tr) I Fn(t) I 2t2dt, m = -I 0, I , R(r) = 4n: [j0 (2n:tr)IF(t)i2t2dt, M(r) = 4n: [j1 (2n:tr)F12(t)t2dt, the correlation tensor having the matrix form M R ( 4.13) ( 4.14) (4.15) An example of correlated vector and scalar random fields is given by the velocity and the pressure fields of atmosphere. Degenerated examples are provided by I 1 =grad I, 12 =1 and 11 =1, l 2 =div l; the former case was discussed in I, §6.1. as potential random field. The corresponding spectral density tensors have the forms respectively 0 I 2n:tF 1 2 0 2n:t IF I 2 -2n:t I F0 I 2 I 2n:tF0 I 2 5. Derivation of the spectral representation We denote k= (211 + 1) + (2l2+ I) in the following. In this paper we derive the spectral representation (3.9) from a k-dimensional moving average (2.13), where p=k as mentioned at the end of section 2. The procedure essentially follows that of I; the argument used for a vector is now applied to a 11 +l2-vector. Hence we describe the procedure briefly. The spectral density tensor (2.18) is an isotropic tensor field having the form (3.4). Therefore, we must find k V0 -valued (k-dimensional) vector functions {F':,(y), a= I,···, k} m= -J., ···, J., J.=li, 12 , which satisfy Simultaneous Spectral Representations of lsotropicaUy Correlated Random Fields 573 (5.1) m=-J., ... ,J., n=-µ,, .. ,µ. Let us regard Va as another representation space D1, +D12 , and divide the vector components {F~"', a=l, , .. , k} into the canonical components belonging to either l,- or /2-vector, which we write as F~"'(y) = cpl'-~ (y), µ = l1 , l2 , a= -µ, ... , µ, "'"' (5.2) where the superscript a denoting a covariant component is lowered and dotted. Let VR stand for the k-dimensional vector space with respect to the subscript m. Furthermore, we regard cp~I'-, with respect to the subscripts m, a and the super-"'"' scripts )., µ, as an isotropic tensor field in the product space VR X Va= (D11 +D12 ) X (D1 1+D12): (5.3) Then we find that ( 5.2) satisfies ( 5.1) with (5.4) n = -L, ... , L, where L denotes the smaller of). and µ, and that F~P.(t) is a nonnegative definite Hermitian matrix with respect to the indices ). and µ. Now that cp>.P:. (y), "'"' namely F;::'(y), is an isotropic tensor field, its Fourier transformf~(x) is another isotropic tensor field having the representation9 (5.5) J;:(r) = 4n-i>.-µ•tL f p;,.~(2n-tr)cp~1-'(t)t 2dt, ). , µ = l" l2 , m = - L, · .. , L . (5.6) To supress the trivial factor i>.-p,, we replace i>.-p,cp>.p, by cp>..p, and i>..-p,p>..P. by p>.P. in what follows. The Hermitian property of p>..p, is never destroyed by doing so. In order to integrate the moving average (2.13), we put p = x-x', P = I x-x' I = v r2 +r'2 -2rr2 cos e, (5.7) where 8 is the angle between x and x ', and apply the tensor addition formula 9 See II, Appendix II. 574 Hisanao 0GuRA II, (4.1) to the tensor function (5.5): = (4n-) 2 ~ ~ ~ f00 J/t)n(2n-tr, 0, cp)J~;m(2n-tr', 0, <p 1)<p~"(t)t 2dt n l B Jo = v4n;,1JL ~.t,[1i:m(2n-tr, {}, <p)dEf:m (t).F'", where we have put dE/;l,.(t) = Ef~m(dt) and Efi:(A)P;,; = (4n-) 312 L Jf;l,.(2n-tr, 0, <p)<p~"(t)t 2dt, (5.8) (5.9) for any interval A on T. Eq. (5.9) is a µ-vector function and we consider a sum-vector with µ=li, [2 , namely, a l1 +l2-vector function in the k-dimensional vector space Va, Then the inner product of two such Va-valued vector functions becomes (EU1)~(A).f_-'1+EU2)n(A)j>'12, EUS,.,(A')J"11 +EU;{n,(A')J"12) ( 5.10) by virtue of II, (3.8). The k-dimensional stochastic integral of the V,i-valued function10 then gives a random measure on T having properties E(M/:)n(A)M/~:;,(A')) = onn'o"'os/4n- ~ f 1<p~v(t)<p~v(t)t 2dt, V=/1'/2 ] t. nt. :x,, µ = l1, l1 ' (5.12) by virtue of (2.11) and (5.10). Consequently, ifwe make a direct-sum of [1 - and [ 2-vectors corresponding to JL=l1 , l2 on both the sides of (5.8) and take the k-di-mentional stochastic integral of l1 +!2-vector function termwise, then we obtain the spectral representations (3.9). Acknowledgement The author is gratefully indebted to Professor J. Ikenoue for his advice and encouragement during this research. 10 The k-dimensional stochastic integral as defined by (2.8) is easily modified so as to apply to a Vu-valued function expressed in terms of canonical component. See I, §5.1, 5.2. Simultaneous Spectral Representations of Isotropically Correlated Random Fields 575 References 1) H. Ogura: J. Phys. Soc.Japan. 21, 154 (1966) 2) H. Ogura: J. Phys. Soc.Japan. 24, 1370 (1968) 3) J.M. Gel'fand, R.A. Minlos and Z.Ya. Shapiro: Representations of the Rotation and Lorenz Group and Their Applications, Pergamon Press, 1963 4) H. Ogura: J. Phys. Soc.Japan. 25,586 (1968) 5) H. Ognra: J. Phys. Soc. Japan 27, 272 (1969) 

Simultaneous Spectral Representations of Isotropically Correlated Random Fields

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Simultaneous Spectral Representations of Isotropically Correlated Random Fields

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